Boundary Hyperbolic Packets in Axisymmetric Euler Flow

Abstract

We study hyperbolic side-wall packets for the axisymmetric Euler equations with swirl in a periodic cylinder. In the exact odd class, $\Gamma =r{u}^{\theta }$ and $G={\omega }^{\theta }/r$ satisfy $D_t\Gamma =0$ and $D_{t}G=r^{-4}{\partial }_z({\Gamma }^2)$, and the symmetry fixes the boundary point at which the relevant swirl and vorticity gradients are measured. We derive the five-dimensional side-wall Green-kernel expansion, its leading signed hyperbolic kernel, packet--core coupling, off-diagonal shear estimates, anisotropy and source-curvature identities, and a maximal conic-score formulation. Under a common admissible packet bootstrap these estimates imply the Dini system $D^{+}\mathfrak{M}\ge c{B}^2$, $B^{\prime }\ge c\mathfrak{M}B$, and hence finite-time blow-up of the comparison amplitudes. The result is a conditional amplification criterion: it does not prove that one smooth Euler trajectory preserves all packet-dominance, tail, shear, anisotropy, and validity hypotheses until the comparison blow-up time. Establishing or refuting that persistence is the remaining problem for this mechanism.